yeah by default, it "should" sync with your google account. that is, if you are using one (play store is enough - if you can download apps, you probably have a google account).
login to gmail and check your contacts.
if not, its bad news - unless they were stored on your SIM (which is unlikely unless they are old contacts) then you probably will not see them again. there would be no recovery software that i could think of that could bring them back from your internal flash media (not SD card).
might be a nasty lesson i guess.....good luck!!
EDIT - just re-read your first post, using your sim for an export is not a good idea. sims generally use an older way of storing your contacts, and you will most probably lose info (as it seems you have). they cannot deal with multi-entry contacts - such as name, mobile, work, email, etc in one contact. this will break.
you can export to SDCARD generally, or sync to gmail as previously mentioned. hope this might help someone else one day
there are free apps such as My Backup or Titanium Backup that will handle this for you perfectly, even save your SMS and MMS into a handy ZIP file. Sometimes this requires root access but you will get basic features on a stock phone.
XDA Developers has an awesome forum, with some very clever people on it. have a search on there, see if anyone has a hack way to recover data such as your loss.
